2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 85.8%

Time: 39.2s

Alternatives: 18

Speedup: 1.9×

• could not determine a ground truth

  1. x = -2
  2. n = 4.94066e-324

Specification

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
end function

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
end function

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{\left(\frac{0.6666666666666666}{n}\right)}, \sqrt[3]{-t_0}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (/ (/ 1.0 x) n)
         (fma
          (pow x (/ 0.6666666666666666 n))
          (cbrt (- t_0))
          (exp (/ (log1p x) n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = fma(pow(x, (0.6666666666666666 / n)), cbrt(-t_0), exp((log1p(x) / n)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = fma((x ^ Float64(0.6666666666666666 / n)), cbrt(Float64(-t_0)), exp(Float64(log1p(x) / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(0.6666666666666666 / n), $MachinePrecision]], $MachinePrecision] * N[Power[(-t$95$0), 1/3], $MachinePrecision] + N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 94.6%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 94.6%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 99.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-rgt-identity 99.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      7. exp-to-pow 99.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999973e-21 < (/.f64 1 n)

   1. Initial program 58.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg 58.4%

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left(-{x}^{\left(\frac{1}{n}\right)}\right)} \]

      2. +-commutative 58.4%

         \[\leadsto \color{blue}{\left(-{x}^{\left(\frac{1}{n}\right)}\right) + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \]

      3. add-cube-cbrt 58.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \]

      4. fma-def 58.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      5. add-exp-log 58.4%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}}\right) \]

      6. log-pow 58.4%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}}\right) \]

      7. +-commutative 58.4%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}}\right) \]

      8. log1p-udef 95.4%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}}\right) \]

      9. *-commutative 95.4%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}}\right) \]

      10. un-div-inv 95.4%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right) \]

   4. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 95.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-udef 95.3%

         \[\leadsto \mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}\right)} - 1}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      3. cbrt-unprod 95.5%

         \[\leadsto \mathsf{fma}\left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)}}\right)} - 1, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      4. sqr-neg 95.5%

         \[\leadsto \mathsf{fma}\left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}}\right)} - 1, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      5. pow-sqr 95.5%

         \[\leadsto \mathsf{fma}\left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{{x}^{\left(2 \cdot \frac{1}{n}\right)}}}\right)} - 1, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      6. div-inv 95.5%

         \[\leadsto \mathsf{fma}\left(e^{\mathsf{log1p}\left(\sqrt[3]{{x}^{\color{blue}{\left(\frac{2}{n}\right)}}}\right)} - 1, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

   6. Applied egg-rr 95.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{x}^{\left(\frac{2}{n}\right)}}\right)} - 1}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]
   7. Step-by-step derivation

      1. expm1-def 95.3%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{x}^{\left(\frac{2}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-log1p 95.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{{x}^{\left(\frac{2}{n}\right)}}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

   8. Simplified 95.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{{x}^{\left(\frac{2}{n}\right)}}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]
   9. Step-by-step derivation

      1. pow1/3 95.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({x}^{\left(\frac{2}{n}\right)}\right)}^{0.3333333333333333}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. pow-pow 95.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{2}{n} \cdot 0.3333333333333333\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

   10. Applied egg-rr 95.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{2}{n} \cdot 0.3333333333333333\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]
   11. Step-by-step derivation

       1. associate-*l/ 95.5%

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{\left(\frac{2 \cdot 0.3333333333333333}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

       2. metadata-eval 95.5%

          \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{\color{blue}{0.6666666666666666}}{n}\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

   12. Simplified 95.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{0.6666666666666666}{n}\right)}}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{\left(\frac{0.6666666666666666}{n}\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (/ (/ 1.0 x) n)
         (exp (log (- (exp (/ (log1p x) n)) t_0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = exp(log((exp((log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = Math.exp(Math.log((Math.exp((Math.log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	else:
		tmp = math.exp(math.log((math.exp((math.log1p(x) / n)) - t_0)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = exp(log(Float64(exp(Float64(log1p(x) / n)) - t_0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[Exp[N[Log[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 94.6%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 94.6%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 99.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-rgt-identity 99.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      7. exp-to-pow 99.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999973e-21 < (/.f64 1 n)

   1. Initial program 58.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 58.5%

         \[\leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}} \]

      2. add-exp-log 58.5%

         \[\leadsto e^{\log \left(\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}\right)} \]

      3. log-pow 58.5%

         \[\leadsto e^{\log \left(e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}\right)} \]

      4. +-commutative 58.5%

         \[\leadsto e^{\log \left(e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}\right)} \]

      5. log1p-udef 95.4%

         \[\leadsto e^{\log \left(e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}\right)} \]

      6. *-commutative 95.4%

         \[\leadsto e^{\log \left(e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\right)} \]

      7. un-div-inv 95.4%

         \[\leadsto e^{\log \left(e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\right)} \]

   4. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{\mathsf{log1p}\left(x\right) \cdot \frac{2}{n}}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (/ (/ 1.0 x) n)
         (- (sqrt (exp (* (log1p x) (/ 2.0 n)))) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = sqrt(exp((log1p(x) * (2.0 / n)))) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = Math.sqrt(Math.exp((Math.log1p(x) * (2.0 / n)))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	else:
		tmp = math.sqrt(math.exp((math.log1p(x) * (2.0 / n)))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(sqrt(exp(Float64(log1p(x) * Float64(2.0 / n)))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Sqrt[N[Exp[N[(N[Log[1 + x], $MachinePrecision] * N[(2.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 94.6%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 94.6%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 99.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-rgt-identity 99.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      7. exp-to-pow 99.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999973e-21 < (/.f64 1 n)

   1. Initial program 58.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0 58.4%

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   4. Step-by-step derivation

      1. log1p-def 95.4%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Simplified 95.4%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 95.3%

         \[\leadsto \color{blue}{\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} \cdot \sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. sqrt-prod 95.4%

         \[\leadsto \color{blue}{\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} \cdot e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}} - {x}^{\left(\frac{1}{n}\right)} \]

      3. unpow2 95.4%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}}} - {x}^{\left(\frac{1}{n}\right)} \]

      4. pow-exp 95.4%

         \[\leadsto \sqrt{\color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n} \cdot 2}}} - {x}^{\left(\frac{1}{n}\right)} \]

      5. div-inv 95.4%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}\right)} \cdot 2}} - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*l* 95.4%

         \[\leadsto \sqrt{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \left(\frac{1}{n} \cdot 2\right)}}} - {x}^{\left(\frac{1}{n}\right)} \]

      7. associate-/r/ 95.4%

         \[\leadsto \sqrt{e^{\mathsf{log1p}\left(x\right) \cdot \color{blue}{\frac{1}{\frac{n}{2}}}}} - {x}^{\left(\frac{1}{n}\right)} \]

      8. clear-num 95.4%

         \[\leadsto \sqrt{e^{\mathsf{log1p}\left(x\right) \cdot \color{blue}{\frac{2}{n}}}} - {x}^{\left(\frac{1}{n}\right)} \]

   7. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\sqrt{e^{\mathsf{log1p}\left(x\right) \cdot \frac{2}{n}}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{\mathsf{log1p}\left(x\right) \cdot \frac{2}{n}}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (/ (/ 1.0 x) n)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 94.6%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 94.6%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 99.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-rgt-identity 99.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      7. exp-to-pow 99.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999973e-21 < (/.f64 1 n)

   1. Initial program 58.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0 58.4%

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   4. Step-by-step derivation

      1. log1p-def 95.4%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Simplified 95.4%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{1 + x}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 4e+149)
           (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)
           (log (pow (/ (+ 1.0 x) x) (/ 1.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e+149) {
		tmp = pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = log(pow(((1.0 + x) / x), (1.0 / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e+149) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = Math.log(Math.pow(((1.0 + x) / x), (1.0 / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 4e+149:
		tmp = math.pow((1.0 + x), (1.0 / n)) - t_0
	else:
		tmp = math.log(math.pow(((1.0 + x) / x), (1.0 / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 4e+149)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0);
	else
		tmp = log((Float64(Float64(1.0 + x) / x) ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+149], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[N[Power[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 94.6%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 94.6%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 99.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-rgt-identity 99.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      7. exp-to-pow 99.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.0000000000000002e149

   1. Initial program 77.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   if 4.0000000000000002e149 < (/.f64 1 n)

   1. Initial program 25.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 6.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 6.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. add-log-exp 78.5%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}\right)} \]

      2. div-inv 78.5%

         \[\leadsto \log \left(e^{\color{blue}{\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}}}\right) \]

      3. exp-prod 78.5%

         \[\leadsto \log \color{blue}{\left({\left(e^{\mathsf{log1p}\left(x\right) - \log x}\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      4. exp-diff 78.5%

         \[\leadsto \log \left({\color{blue}{\left(\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}\right)}}^{\left(\frac{1}{n}\right)}\right) \]

      5. add-exp-log 78.5%

         \[\leadsto \log \left({\left(\frac{e^{\mathsf{log1p}\left(x\right)}}{\color{blue}{x}}\right)}^{\left(\frac{1}{n}\right)}\right) \]

      6. log1p-udef 78.5%

         \[\leadsto \log \left({\left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}^{\left(\frac{1}{n}\right)}\right) \]

      7. rem-exp-log 78.5%

         \[\leadsto \log \left({\left(\frac{\color{blue}{1 + x}}{x}\right)}^{\left(\frac{1}{n}\right)}\right) \]

      8. +-commutative 78.5%

         \[\leadsto \log \left({\left(\frac{\color{blue}{x + 1}}{x}\right)}^{\left(\frac{1}{n}\right)}\right) \]

   7. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{\log \left({\left(\frac{x + 1}{x}\right)}^{\left(\frac{1}{n}\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{1 + x}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{1 + x}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 4e+149)
           (- (+ 1.0 (/ x n)) t_0)
           (log (pow (/ (+ 1.0 x) x) (/ 1.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e+149) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log(pow(((1.0 + x) / x), (1.0 / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e+149) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log(Math.pow(((1.0 + x) / x), (1.0 / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 4e+149:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log(math.pow(((1.0 + x) / x), (1.0 / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 4e+149)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log((Float64(Float64(1.0 + x) / x) ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+149], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[N[Power[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 94.6%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 94.6%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 99.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-rgt-identity 99.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      7. exp-to-pow 99.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.0000000000000002e149

   1. Initial program 77.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 74.3%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if 4.0000000000000002e149 < (/.f64 1 n)

   1. Initial program 25.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 6.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 6.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. add-log-exp 78.5%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}\right)} \]

      2. div-inv 78.5%

         \[\leadsto \log \left(e^{\color{blue}{\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}}}\right) \]

      3. exp-prod 78.5%

         \[\leadsto \log \color{blue}{\left({\left(e^{\mathsf{log1p}\left(x\right) - \log x}\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      4. exp-diff 78.5%

         \[\leadsto \log \left({\color{blue}{\left(\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}\right)}}^{\left(\frac{1}{n}\right)}\right) \]

      5. add-exp-log 78.5%

         \[\leadsto \log \left({\left(\frac{e^{\mathsf{log1p}\left(x\right)}}{\color{blue}{x}}\right)}^{\left(\frac{1}{n}\right)}\right) \]

      6. log1p-udef 78.5%

         \[\leadsto \log \left({\left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}^{\left(\frac{1}{n}\right)}\right) \]

      7. rem-exp-log 78.5%

         \[\leadsto \log \left({\left(\frac{\color{blue}{1 + x}}{x}\right)}^{\left(\frac{1}{n}\right)}\right) \]

      8. +-commutative 78.5%

         \[\leadsto \log \left({\left(\frac{\color{blue}{x + 1}}{x}\right)}^{\left(\frac{1}{n}\right)}\right) \]

   7. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{\log \left({\left(\frac{x + 1}{x}\right)}^{\left(\frac{1}{n}\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{1 + x}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e+193)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e+193:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e+193)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 94.6%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 94.6%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 99.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-rgt-identity 99.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      7. exp-to-pow 99.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 70.9%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 6.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 6.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 6.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 74.6%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -5e+214)
     0.0
     (if (<= (/ 1.0 n) -5000.0)
       t_1
       (if (<= (/ 1.0 n) -2e-134)
         (/ (- x (log x)) n)
         (if (<= (/ 1.0 n) -5e-251)
           t_0
           (if (<= (/ 1.0 n) 2e-69)
             (/ (- (log x)) n)
             (if (<= (/ 1.0 n) 5e-21)
               t_0
               (if (<= (/ 1.0 n) 5e+193) t_1 (/ 1.0 (* n x)))))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = 0.0;
	} else if ((1.0 / n) <= -5000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e-134) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= -5e-251) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / x) / n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-5d+214)) then
        tmp = 0.0d0
    else if ((1.0d0 / n) <= (-5000.0d0)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-2d-134)) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= (-5d-251)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-69) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 5d-21) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+193) then
        tmp = t_1
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = 0.0;
	} else if ((1.0 / n) <= -5000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e-134) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= -5e-251) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e+214:
		tmp = 0.0
	elif (1.0 / n) <= -5000.0:
		tmp = t_1
	elif (1.0 / n) <= -2e-134:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= -5e-251:
		tmp = t_0
	elif (1.0 / n) <= 2e-69:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 5e-21:
		tmp = t_0
	elif (1.0 / n) <= 5e+193:
		tmp = t_1
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+214)
		tmp = 0.0;
	elseif (Float64(1.0 / n) <= -5000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -2e-134)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= -5e-251)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+193)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -5e+214)
		tmp = 0.0;
	elseif ((1.0 / n) <= -5000.0)
		tmp = t_1;
	elseif ((1.0 / n) <= -2e-134)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= -5e-251)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-69)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 5e-21)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+193)
		tmp = t_1;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+214], 0.0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-134], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-251], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], t$95$1, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -4.99999999999999953e214

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 100.0%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 100.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 100.0%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 100.0%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 100.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 100.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around 0 32.3%

      \[\leadsto \log \left(e^{\color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}}}\right) \]

   6. Taylor expanded in n around inf 69.2%

      \[\leadsto \log \color{blue}{1} \]

   if -4.99999999999999953e214 < (/.f64 1 n) < -5e3 or 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 90.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 66.7%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if -5e3 < (/.f64 1 n) < -2.00000000000000008e-134

   1. Initial program 14.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 73.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 73.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around 0 59.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n} + \frac{x}{n}} \]
   7. Step-by-step derivation

      1. neg-mul-1 59.7%

         \[\leadsto \color{blue}{\left(-\frac{\log x}{n}\right)} + \frac{x}{n} \]

      2. +-commutative 59.7%

         \[\leadsto \color{blue}{\frac{x}{n} + \left(-\frac{\log x}{n}\right)} \]

      3. sub-neg 59.7%

         \[\leadsto \color{blue}{\frac{x}{n} - \frac{\log x}{n}} \]

      4. div-sub 59.7%

         \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]

   8. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]

   if -2.00000000000000008e-134 < (/.f64 1 n) < -5.0000000000000003e-251 or 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 38.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 65.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 65.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 73.1%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if -5.0000000000000003e-251 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 30.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 30.3%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in n around inf 64.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ 64.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \log x}{n}} \]

      2. mul-1-neg 64.7%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   6. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 6.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 6.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 6.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 74.6%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e+214)
     t_1
     (if (<= (/ 1.0 n) -5e+29)
       t_0
       (if (<= (/ 1.0 n) 2e-69)
         t_1
         (if (<= (/ 1.0 n) 5e-21)
           (/ (/ 1.0 x) n)
           (if (<= (/ 1.0 n) 5e+193) t_0 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e+29) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d+214)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d+29)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-69) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-21) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d+193) then
        tmp = t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e+29) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e+214:
		tmp = t_1
	elif (1.0 / n) <= -5e+29:
		tmp = t_0
	elif (1.0 / n) <= 2e-69:
		tmp = t_1
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e+193:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+214)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e+29)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e+193)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e+214)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e+29)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-69)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-21)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e+193)
		tmp = t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+214], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+29], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.99999999999999953e214 or -5.0000000000000001e29 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 43.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 80.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 80.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 80.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      2. diff-log 80.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

      3. +-commutative 80.1%

         \[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n} \]

   7. Applied egg-rr 80.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n} \]

   if -4.99999999999999953e214 < (/.f64 1 n) < -5.0000000000000001e29 or 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 89.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.1%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 6.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 6.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 6.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 74.6%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e+193)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-16)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-69) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d-21) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d+193) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e+193:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e+193)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-16)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-69)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e-21)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e+193)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 94.6%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 94.6%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 99.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-rgt-identity 99.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      7. exp-to-pow 99.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 86.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      2. diff-log 86.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

      3. +-commutative 86.1%

         \[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n} \]

   7. Applied egg-rr 86.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 70.9%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 6.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 6.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 6.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 74.6%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e+193) (- 1.0 t_0) (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-16)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-69) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d-21) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d+193) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e+193:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e+193)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-16)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-69)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e-21)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e+193)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 94.6%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 94.6%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 94.6%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 94.6%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 99.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-rgt-identity 99.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      7. exp-to-pow 99.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 86.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      2. diff-log 86.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

      3. +-commutative 86.1%

         \[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n} \]

   7. Applied egg-rr 86.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 70.8%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 6.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 6.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 6.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 74.6%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+57}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- x (log x)) n)
   (if (<= x 7.5e+57) 0.0 (if (<= x 1.9e+183) (/ (/ 1.0 x) n) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 7.5e+57) {
		tmp = 0.0;
	} else if (x <= 1.9e+183) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 7.5d+57) then
        tmp = 0.0d0
    else if (x <= 1.9d+183) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 7.5e+57) {
		tmp = 0.0;
	} else if (x <= 1.9e+183) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 7.5e+57:
		tmp = 0.0
	elif x <= 1.9e+183:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 7.5e+57)
		tmp = 0.0;
	elseif (x <= 1.9e+183)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 7.5e+57)
		tmp = 0.0;
	elseif (x <= 1.9e+183)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 7.5e+57], 0.0, If[LessEqual[x, 1.9e+183], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 44.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 49.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 49.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around 0 48.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n} + \frac{x}{n}} \]
   7. Step-by-step derivation

      1. neg-mul-1 48.2%

         \[\leadsto \color{blue}{\left(-\frac{\log x}{n}\right)} + \frac{x}{n} \]

      2. +-commutative 48.2%

         \[\leadsto \color{blue}{\frac{x}{n} + \left(-\frac{\log x}{n}\right)} \]

      3. sub-neg 48.2%

         \[\leadsto \color{blue}{\frac{x}{n} - \frac{\log x}{n}} \]

      4. div-sub 48.2%

         \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]

   8. Simplified 48.2%

      \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]

   if 1 < x < 7.5000000000000006e57 or 1.9e183 < x

   1. Initial program 81.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 81.8%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 81.8%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 81.8%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 81.8%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 81.8%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 81.8%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 81.8%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around 0 13.5%

      \[\leadsto \log \left(e^{\color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}}}\right) \]

   6. Taylor expanded in n around inf 81.8%

      \[\leadsto \log \color{blue}{1} \]

   if 7.5000000000000006e57 < x < 1.9e183

   1. Initial program 50.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 50.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 50.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 74.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+57}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+59}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- (log x)) n)
   (if (<= x 2e+59) 0.0 (if (<= x 1.25e+181) (/ (/ 1.0 x) n) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = -log(x) / n;
	} else if (x <= 2e+59) {
		tmp = 0.0;
	} else if (x <= 1.25e+181) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = -log(x) / n
    else if (x <= 2d+59) then
        tmp = 0.0d0
    else if (x <= 1.25d+181) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2e+59) {
		tmp = 0.0;
	} else if (x <= 1.25e+181) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = -math.log(x) / n
	elif x <= 2e+59:
		tmp = 0.0
	elif x <= 1.25e+181:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2e+59)
		tmp = 0.0;
	elseif (x <= 1.25e+181)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -log(x) / n;
	elseif (x <= 2e+59)
		tmp = 0.0;
	elseif (x <= 1.25e+181)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2e+59], 0.0, If[LessEqual[x, 1.25e+181], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 44.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 42.7%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in n around inf 47.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ 47.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \log x}{n}} \]

      2. mul-1-neg 47.8%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   6. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   if 1 < x < 1.99999999999999994e59 or 1.2500000000000001e181 < x

   1. Initial program 81.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 81.8%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 81.8%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 81.8%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 81.8%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 81.8%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 81.8%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 81.8%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around 0 13.5%

      \[\leadsto \log \left(e^{\color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}}}\right) \]

   6. Taylor expanded in n around inf 81.8%

      \[\leadsto \log \color{blue}{1} \]

   if 1.99999999999999994e59 < x < 1.2500000000000001e181

   1. Initial program 50.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 50.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 50.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 74.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+59}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.1% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000000000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -10000000000000.0) 0.0 (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000000000000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-10000000000000.0d0)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000000000000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -10000000000000.0:
		tmp = 0.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10000000000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -10000000000000.0)
		tmp = 0.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -10000000000000.0], 0.0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -1e13

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-exp-log 100.0%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      3. log-pow 100.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      4. +-commutative 100.0%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      5. log1p-udef 100.0%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 100.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 100.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around 0 54.6%

      \[\leadsto \log \left(e^{\color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}}}\right) \]

   6. Taylor expanded in n around inf 47.0%

      \[\leadsto \log \color{blue}{1} \]

   if -1e13 < (/.f64 1 n)

   1. Initial program 38.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 60.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 60.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 43.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000000000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.9% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 1.0 / (n * x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 1.0d0 / (n * x)
end function

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 56.2%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-def 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 38.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
7. Step-by-step derivation

   1. *-commutative 38.5%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

8. Simplified 38.5%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

9. Final simplification 38.5%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 16: 40.5% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (1.0 / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 56.2%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-def 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 38.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
7. Step-by-step derivation

   1. associate-/r* 39.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

8. Simplified 39.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

9. Final simplification 39.2%

   \[\leadsto \frac{\frac{1}{n}}{x} \]
10. Add Preprocessing

Alternative 17: 40.5% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 x) n))

C

double code(double x, double n) {
	return (1.0 / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / x) / n
end function

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

function code(x, n)
	return Float64(Float64(1.0 / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 56.2%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-def 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 39.2%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

7. Final simplification 39.2%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
8. Add Preprocessing

Alternative 18: 4.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

public static double code(double x, double n) {
	return x / n;
}

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

function tmp = code(x, n)
	tmp = x / n;
end

Wolfram

code[x_, n_] := N[(x / n), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 54.2%

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   2. add-exp-log 54.2%

      \[\leadsto \log \left(e^{\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   3. log-pow 54.2%

      \[\leadsto \log \left(e^{e^{\color{blue}{\frac{1}{n} \cdot \log \left(x + 1\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. +-commutative 54.2%

      \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \log \color{blue}{\left(1 + x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   5. log1p-udef 60.5%

      \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   6. *-commutative 60.5%

      \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   7. un-div-inv 60.5%

      \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

4. Applied egg-rr 60.5%

   \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

5. Taylor expanded in x around 0 40.7%

   \[\leadsto \log \left(e^{\color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}}}\right) \]

6. Taylor expanded in x around inf 4.5%

   \[\leadsto \color{blue}{\frac{x}{n}} \]

7. Final simplification 4.5%

   \[\leadsto \frac{x}{n} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))